Deterministic Discrimination of Phase-Modified Permutation Oracles via Single Qubit Measurement

TL;DR

This paper presents a quantum method to distinguish between two types of permutation oracles with a single query and minimal measurement, exploiting quantum phase differences without ancilla qubits.

Contribution

It introduces a novel, efficient quantum discrimination protocol for phase-modified permutation oracles using only single-qubit measurement and no ancilla qubits.

Findings

Single-query, one-qubit measurement distinguishes cases with certainty

Protocol requires only $n+1$ Hadamard gates and no ancilla qubits

The method exploits intrinsic quantum phase differences

Abstract

I study a promise problem for an unknown unitary operator acting on an $n$-qubit system. The operator is promised to take one of two forms: either it implements a fixed permutation of computational basis states, or it implements the same permutation together with a conditional sign change determined by a designated input qubit. I show that these two cases can be distinguished with certainty using a single query to the unknown operator and a measurement of only one qubit. The procedure requires no ancilla qubits and uses only $n+1$ Hadamard gates in addition to the oracle call. The promise is intrinsically quantum, since the two cases differ only in their relative-phase structure and therefore have no direct classical counterpart in the usual black-box model.